Our focus is on \(\theta _{\rm max}({{\mathcal {I}}}^{\Lambda ,\tau }_{G})\) , i.e., the maximum eigenvalue of
\({{\mathcal {I}}}^{\Lambda ,\tau }_{G}\) . If for any choice of \(\Lambda , \tau \)  we have that \(\theta _{\rm max}({{\mathcal {I}}}^{\Lambda ,\tau }_{G})=O(1)\) ,
then we say that the Gibbs distribution \(\mu \)  exhibits spectral independence. Hence, the name of the method.
Spectral independence for \(\mu \)  implies that the corresponding Glauber dynamics has polynomial mixing time
[1]}. The precise magnitude of the mixing time in this case is a subject of intense study. Here we don't focus
on this problem, i.e., we focus only on establishing spectral independence for \(\mu \) .
